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Properties

Label 2-608-19.18-c2-0-19
Degree $2$
Conductor $608$
Sign $0.719 - 0.694i$
Analytic cond. $16.5668$
Root an. cond. $4.07023$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.94i·3-s − 7.51·5-s + 11.4·7-s − 6.58·9-s + 14.4·11-s − 23.9i·13-s − 29.6i·15-s + 26.8·17-s + (−13.2 − 13.6i)19-s + 45.1i·21-s − 0.456·23-s + 31.4·25-s + 9.52i·27-s + 7.85i·29-s − 16.0i·31-s + ⋯
L(s)  = 1  + 1.31i·3-s − 1.50·5-s + 1.63·7-s − 0.731·9-s + 1.31·11-s − 1.84i·13-s − 1.97i·15-s + 1.57·17-s + (−0.694 − 0.719i)19-s + 2.14i·21-s − 0.0198·23-s + 1.25·25-s + 0.352i·27-s + 0.270i·29-s − 0.519i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(16.5668\)
Root analytic conductor: \(4.07023\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1),\ 0.719 - 0.694i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.834628609\)
\(L(\frac12)\) \(\approx\) \(1.834628609\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (13.2 + 13.6i)T \)
good3 \( 1 - 3.94iT - 9T^{2} \)
5 \( 1 + 7.51T + 25T^{2} \)
7 \( 1 - 11.4T + 49T^{2} \)
11 \( 1 - 14.4T + 121T^{2} \)
13 \( 1 + 23.9iT - 169T^{2} \)
17 \( 1 - 26.8T + 289T^{2} \)
23 \( 1 + 0.456T + 529T^{2} \)
29 \( 1 - 7.85iT - 841T^{2} \)
31 \( 1 + 16.0iT - 961T^{2} \)
37 \( 1 - 12.5iT - 1.36e3T^{2} \)
41 \( 1 - 48.5iT - 1.68e3T^{2} \)
43 \( 1 - 30.5T + 1.84e3T^{2} \)
47 \( 1 - 44.3T + 2.20e3T^{2} \)
53 \( 1 + 48.7iT - 2.80e3T^{2} \)
59 \( 1 + 43.7iT - 3.48e3T^{2} \)
61 \( 1 - 3.64T + 3.72e3T^{2} \)
67 \( 1 - 75.4iT - 4.48e3T^{2} \)
71 \( 1 + 19.4iT - 5.04e3T^{2} \)
73 \( 1 + 83.0T + 5.32e3T^{2} \)
79 \( 1 - 73.9iT - 6.24e3T^{2} \)
83 \( 1 + 4.27T + 6.88e3T^{2} \)
89 \( 1 - 105. iT - 7.92e3T^{2} \)
97 \( 1 + 177. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.69240328859136700178576127111, −9.804757912825795380809482500043, −8.626462222073228345270099575979, −8.076657362134188182279155344801, −7.32044433897912342344058787642, −5.61317523364956733896572728461, −4.72531907741087706316586170180, −4.04719130823081290342768627679, −3.19881100853142855324595099050, −0.965401911353103383095344968937, 1.10094263286726530325159124220, 1.89694625859548104623187341199, 3.86624106833007416687306915369, 4.47068372879399678801317901211, 6.00180577113456859874766496416, 7.13697688333390647855155351458, 7.54560351203863894714322316130, 8.338041598147359017721336650094, 9.063434507296137413337679904451, 10.70020244917540630861478332274

Graph of the $Z$-function along the critical line